Remark 2.23 
In the proof of Proposition REF  we rectified the fast foliation in a neighbourhood of a normally hyperbolic point of \(S\) . By a subsequent application of the coordinate transformation used in the proof of Proposition REF , one can also rectify the critical manifold \(S\)  in such neighbourhoods in order to obtain a local normal form analogous to the well-known (local) Fenichel normal form [1]}, [2]}, [3]} in the continuous-time setting.

